The strong law of \(U\)-statistics with \(\varphi\)-mixing samples
Let \(\{X_n : n \in \mathbb{Z}\}\) be a strictly stationary process on a probability space \((\Omega, {\mathcal A}, P)\). Denote by \(F\) the distribution function of \(X_1\) and by \({\mathcal I}^b_a\) the \(\sigma\)-algebra generated by \(X_a, \ldots, X_b\) \((a \leq b)\). For each \(n\in \mathbb{Z}\), let \[ \varphi^* (n) = \sup \Bigl\{ \max \biggl( \bigl |P (B \mid A) - P(B) \bigr |,\;\bigl |P(A \mid B) - P(A) \bigr |\biggr) : A \in {\mathcal I}^a_{- \infty},\;B \in {\mathcal I}^\infty_{a + n},\;a \in \mathbb{Z} \Bigr\}. \] The process \(\{X_n\}\) is said uniformly mixing in both directions of times \((\varphi^*\)-mixing) if \(\varphi^* (n) \to 0\) as \(n \to \infty\). Let \(h(n,y)\) be a real-valued Borel measurable function, symmetric in its arguments. Define a \(U\)-statistic by \(U_n = {n \choose 2}^{-1} \sum_{1 \leq i < j \leq n} h(X_i, X_j)\). The author obtains the following theorem, which is similar to that in the i.i.d. case, a former result by \textit{R. H. Berk} [Ann. Math. Stat. 37, 51-58 (1966; Zbl 0151.238)]: Suppose that \(\{X_n : n \in \mathbb{Z}\}\) is \(\varphi^*\)-mixing with coefficients \(\varphi^* (n)\) satisfying the condition \(\varphi^* (n) = O(n^{- (4 + \delta)})\) for some \(\delta > 0\), and also assume that \(\sup_{m \geq 2} E |h(X_1,X_m) |< \infty\), \(\iint^\infty_{- \infty} h(x,y) dF(x) dF(y) = 0\). Then \(U_n \to 0\) almost surely, as \(n \to \infty\).
uniformly mixing in both directions of times
scientific article; zbMATH DE number 768787